Going beyond the Golden Ratio (opens in new tab)

Good question, but I don't know and haven't really done anything substantially related that might even give us a hint.

Hopefully someone else chime in on this thread. ;)

What I find fascinating is that there seem to be so many valid ways to generalize the Golden Ratio.

As you say, the "metallic means" [1] are quite well-known, and relate to the recurrence relation via: T(n) = m *T(n-1)+ T(n-2), for some constant integer m. For example, m=1 is the golden ratio, m=2 is the silver ratio,...

But one of my other posts [2], generalizes the Golden ratio via the "Harmonious Numbers", as defined by the lagged recurrence, T(n+m) = T(n)+T(n-1), for some constant m. In this case, m=1 relates to the Golden Ratio, and m=2 relates to the Plastic Number [3].

And then finally, this post explores generalizing it via a completely different perspective, that of "Lagrange Numbers".

It seems that we need to 'think outside the box' a litte when generalizing the Golden ratio, as there is not single obvious way to generalise continued fractions.

[1] https://en.wikipedia.org/wiki/Metallic_mean

[2] http://extremelearning.com.au/unreasonable-effectiveness-of-...

[3] https://en.wikipedia.org/wiki/Plastic_number

Yes.

Consider π. For any S>0, you can construct an infinite number of rational approximations that have a score of less than S.

But for any quadratic irrational (surd), as the depth of the corresponding continued fraction increases, the score will converge (in an alternating manner) to a critical score, S.

This means that for any score S < S, there is only a finite set of rational approximations that have a score of less than S.

For example, in figure 3, for S=0.4 < 1/√5 ≃ 0.447, there is only one fraction that gives a score of less than S=0.4.

Hope that helps!

Thanks. No, you aren't crazy, but maybe my typos made you crazy! ;)

I have now fixed a couple of typos in the grammar and continued fractions expressions for that section.

Generally my answer is that this is for the same reason that fitting lines of best fit to data is nearly always done via a least-squares fitting.

Squaring has a few major benefits.

The first is that is never negative.

Therefore, one might ask why don't we just take absolute value (1-norm)? It turns out that the absolute function makes many calculus expressions very messy. Thus, ironically, when analysing these concepts theoreticlly/algebraically it is usually easier to square the errors (use the 2-norm), rather than the 1-norm.

The x^2 function is a very elegant function that smoothly curves. The |x| function has a pointy corner at x=0, which causes many analytical headaches.

(Although, I must admit that in recent years with large-scale computing, errors based on the absolute value are making a notable comeback, especially in machine learning!)

Secondly, history seems to have shown that squaring is frequently the simplest transformation that leads to non-trivial results. Thus, the principle of Occam's razor, would suggest that 2 is a very good place to begin and end.

Finally, if we consider higher powers, it makes sense to ensure our errors are not negative, so that generally rules out cubes. Finding square roots, and roots of quadratic equations is relatively simple, but finding roots of degree 4 polynomials is very tough, and finding roots of higher even degree polynomials is usually intractable.

Hope that helps!

@svat's comment and link may also be helpful in this regard.

Because our brain is a superb pattern matcher. It can see pattern almost everywhere, even when there are none. That has helped our ancestors find animals that can be hunted, and prepare for flooding events timed at certain intervals (12 full moons until the next harvest/sowing season). But that also leads to situations where coincidental patterns are found, where the constituent parts have no casual relationship to one another.

I bet you can find even more patterns, where there are none: http://www.tylervigen.com/spurious-correlations

It’s called ‘coincidence’: you have so many sites (N) to choose from, and there are N²︎ connections between them. To some degree of accuracy you’re going to find ratios between some of these that are ‘close’ to apparently ‘important’ numbers (and there’s plenty of those, and of course integer multiples thereof, which seem to catch just as much attention).

It’s just a numbers game (excuse the pun). It’s just pure numerology. And an overabundance of ratios and constants and multiples thereof to choose from. It would be pretty unlikely that no such coincidental values would turn up.

The distance from Angkor Wat to Giza is: 4754 miles. And the distance from Gize to Nazca is: 4754 x φ miles.

By adding these, you get that the distance from Angkor Wat to Nazca is 4754 (1+φ) miles.

But φ is defined such that 1+φ = φ², [Verify for yourself that 1.61803398875² = 2.61803398875]

So thus, the distance from Ankor Wat to Nazca can also be described as 4754 φ² miles.

I still see 850/10 and 425/5. EDIT: OK, it seems to have been a cache issue as extremelearning supposed.